Aluminum is slower to heat up, fast to dissipate the heat. Copper heats up faster and then holds onto the heat longer. Also, I'd rather have a 1 oz aluminum light than a 2.5 oz copper light. This is an EDC light which will mostly be riding in a pocket to be present for occassional use and not a search and rescue light being used for hours on end.
http://www.tomshardware.com/forum/118364-29-truth-copper-heatsinks-aluminum-ones

A copper light will transfer the heat from the head to the body faster/more efficiently than aluminium, making the head less hot during use than an aluminium light (when used for the same amount of time). That's the pro with copper lights :) so whilst the body will be hotter for longer, there is less risk of burning your fingers/leg since the head is cooler.

@defg, I believe that's not correct. Heat dissipation is proportional to the temperature difference between an object and its surroundings; conductivity is a property of the metal itself (depends on how many free electrons it has, and how free they are). Copper is a better conductor than aluminum, so it will conduct heat away from the LED faster. Having a higher mass, it's true that for a given geometry (a flashlight of a given size, for example), it will retain more total heat, but, divided by its mass, the heat per unit mass will be the same, i.e., the steady-state temperature of the flashlight in the hand will be the same, ultimately, but it will take longer to reach that temperature than a similar aluminum body, (which will have less mass). Dissipation is to some extent dependent on infra-red color ( a "black" body is a better radiator than a "white" body), but copper and aluminum probably have very similar infra-red "color." Much more critical is the geometry of the surface, which in the Olight case is the same for both metals. If you read down in the tomshardware post, you'll find that the initial claim is corrected.